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He was born at Southwick in Sussex. His father, also named John Pell, was from Southwick, and his mother was Mary Holland, from Halden in Kent. He was the second of two sons, and by the age of six he was an orphan, his father dying in 1616 and his mother the following year. John Pell the elder had a fine library, and this proved valuable to the young Pell as he grew up. He was educated at Steyning Grammar School and entered Trinity College, Cambridge, at the age of thirteen.[1] During his university career he became an accomplished linguist, and even before he took his B.A. degree (in 1629) corresponded with Henry Briggs and other mathematicians. He received his M.A. in 1630, and taught in the short-lived Chichester Academy, set up by Samuel Hartlib.[2][3] On 3 July 1632 he married Ithamaria Reginald (also rendered variously as Ithamara or Ithumaria, with the surname Reginolles), sister of Bathsua Makin.[4] They went on to have four sons and four daughters. Ithumaria died in 1661, and some time before 1669 he remarried.

Pell spent much of the 1630s working under Hartlib's influence, on a variety of topics in the area of pedagogy, encyclopedism and pansophy, combinatorics, and the legacy of Trithemius. By 1638 he had formulated a proposal for a universal language.[5] In mathematics, he concentrated on expanding the scope of algebra in the theory of equations, and on mathematical tables.[6] As part of a joint lobbying effort with Hartlib to find himself support to continue as a researcher, he had his short Idea of Mathematics printed in October 1638.[6] The campaign brought interested responses from Johann Moriaen and Marin Mersenne.[7]

In 1646, on the invitation of Frederick Henry, Prince of Orange, Pell accepted a professorship at the new Orange College at Breda, where he taught until 1652. He realised that war between the English and the Dutch was imminent and that he would be in an extremely difficult position in Breda, so returned to England before the outbreak of the First Anglo-Dutch War in July 1652. After his return, Oliver Cromwell appointed Pell to a post teaching mathematics in London.

From 1654 to 1658 Pell acted as Cromwell's political agent in Zurich to the Protestantcantons of Switzerland; he cooperated with Samuel Morland, the English resident at Geneva.[11] Pell was described in Zurich by the English traveller Sir John Reresby in about 1656 as "a strange unknown person, not unsuiting the people he was sent to, nor the master [Cromwell] he came from. They are here so strict in their religion, they suffer not the Venetian ambassador to hear mass in his own house."[12] Cromwell wanted to split the Protestant cantons of Switzerland off to join a Protestant League, with England at its head. However Pell's negotiations were long drawn out and he returned to England to deliver his report only shortly before Cromwell's death. He was unable to report as he waited in vain for an audience with the ailing Cromwell.

A mathematical pupil and disciple in Switzerland, from 1657, was Johann Heinrich Rahn, known as Rhonius.[13] Rahn is credited with the invention of the division sign ÷ (obelus); it has also been attributed to Pell, who taught Rahn a three-column spreadsheet-style technique of tabulation of calculations, and acted as editor for Rahn's 1659 book Teutsche Algebra in which it appeared. This book by Rahn also contained what would become known as the "Pell equation".[14][15]Diophantine equations was a favourite subject with Pell; he lectured on them at Amsterdam. He is now best remembered, if perhaps erroneously, for the indeterminate equation

ax2+1=y2,{\displaystyle ax^{2}+1=y^{2},}

which is known as Pell's equation. This problem was in fact proposed by Pierre de Fermat first to Bernard Frénicle de Bessy, and in 1657 to all mathematicians. Pell's connection with the problem is through Rahn. It consisted of publication of the solutions of John Wallis and Lord Brouncker in his edition of Thomas Branker's Translation of Rhonius's Algebra (1668); added to his earlier editorial contributions, whatever they were, to the 1659 algebra book written by Rahn (i.e. Rhonius).[16] This new edition by Pell of what was essentially Rahn's work included a great deal of additional material on number theory, amounting to a reply to the 1657 book Exercitationes mathematicae by Frans van Schooten. It is also notable for its inclusion of a Table of Incomposits, an early large factor table.[17]

Many of Pell's manuscripts fell into the hands of Richard Busby, master of Westminster School, and afterwards came into the possession of the Royal Society; they are still preserved in nearly forty folio volumes in the British Library, which contain, not only Pell's own memoirs, but much of his correspondence with the mathematicians of his time.

His chief works are:

Astronomical History of Observations of Heavenly Motions and Appearances (1634)

Ecliptica prognostica (1634)

An Idea of Mathematicks (1638)

Controversy with Longomontanus concerning the Quadrature of the Circle (1646?)

A Table of Ten Thousand Square Numbers (fol.; 1672).

The Idea was a short manifesto. It made three suggestions: a mathematical encyclopedia and bibliography; a complete mathematics research library and collection of instruments, with state sponsorship; and a three-volume comprehensive set of mathematical textbooks, able to convey the state of the art to any scholar.[23]

The most recent study of Pell is by Noel Malcolm and Jacqueline Stedall, John Pell (1611-1685) and His Correspondence with Sir Charles Cavendish: The Mental World of an Early Modern Mathematician (Oxford: Oxford University Press, 2005). ISBN0-19-856484-8